Factor the following expression: $x^2 - 8x - 9$
Solution: When we factor a polynomial, we are basically reversing this process of multiplying linear expressions together: $ \begin{eqnarray} (x + a)(x + b) &=& xx &+& xb + ax &+& ab \\ \\ &=& x^2 &+& {(a + b)}x &+& {ab} \end{eqnarray} $ $ \begin{eqnarray} \hphantom{(x + a)(x + b) }&\hphantom{=}&\hphantom{ xx }&\hphantom{+}&\hphantom{ (a + b)x }&\hphantom{+}& \\ &=& x^2 & & {-8}x& & {-9} \end{eqnarray} $ The coefficient on the $x$ term is $-8$ and the constant term is $-9$ , so to reverse the steps above, we need to find two numbers that add up to $-8$ and multiply to $-9$ You can try out different factors of $-9$ to see if you can find two that satisfy both conditions. If you're stuck and can't think of any, you can also rewrite the conditions as a system of equations and try solving for $a$ and $b$ $ {a} + {b} = {-8}$ $ {a} \times {b} = {-9}$ The two numbers $-9$ and $1$ satisfy both conditions: $ {-9} + {1} = {-8} $ $ {-9} \times {1} = {-9} $ So we can factor the expression as: $(x {-9})(x + {1})$